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The nitrogen-vacancy center structure of diamond has attracted widespread attention due to its high sensitivity in quantum precision measurement. In this paper, a coupled phonon field is used to resonantly regulate the atomic spins of the nitrogen-vacancy center for improving the spin transition efficiency. Firstly, the interaction between phonons and lattice energy is analyzed based on the relationship between the wave function and the lattice displacement vector. The spin transition mechanism is investigated based on phonon resonance regulation, and the strain-induced energy transferable phonon-spin interaction coupling excitation model is established. Secondly, the coefficient matrix satisfying Bloch’s theorem is adopted to develop the phonon spectrum model of the first Brillouin zone characteristic region for different axial nitrogen-vacancy centers. Considering the thermal expansion, the thermal balance properties of phonon resonance system are analyzed and its specific heat model is studied based on the Debye model. Finally, the structure optimization model of different axial nitrogen-vacancy centers under the phonon model is built up based on the molecular dynamics simulation software CASTEP and density functional theory for first-principles research. The structural characteristics, phonon characteristics, and thermodynamic properties of nitrogen-vacancy centers are analyzed. The research results show that the evolution of phonon mode depends on the occupation of the nitrogen-vacancy center. A decrease in thermodynamic entropy accompanies the strengthening of the phonon mode. The covalent bond of diamond with nitrogen-vacancy center is weaker than that of a defect-free diamond. The thermodynamic properties of a defect-free diamond are more unstable. The primary phonon resonance frequency of diamond with nitrogen-vacancy centers are on the order of THz, and the secondary phonon resonance frequency is about in a range of 800 and 1200 MHz. A surface acoustic wave resonance mechanism with an interdigital width of 1.5 μm is designed according to the secondary resonance frequency, and its center frequency is about 930 MHz. The phonon resonance control method can effectively increase the spin transition probability of nitrogen-vacancy center under suitable phonon resonance control parameters, and thus realizing the increase of atomic spin manipulation efficiency.
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Keywords:
- nitrogen-vacancy center /
- phonon coupling /
- atomic spin /
- resonance manipulation
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图 1 NV色心几何结构和自旋跃迁性质 (a)几何结构; (b)能级结构及自旋跃迁性质
Figure 1. Structures and spin transition properties of a negatively charged NV center: (a) Geometric structure; (b) energy level structure and spin transition properties.
图 2 NV色心量子化轴示意图
Figure 2. Schematic diagram of quantization axis for NV center
图 3 金刚石中4个轴向NV色心分布及NV坐标系
Figure 3. Four axial NV center distributions and their NV coordinate systems in diamond.
图 4 (a)声子场共振结构示意图; (b)声子场共振调控机理示意图[48]
Figure 4. (a) Schematic diagram of phonon field resonance structure; (b) mechanism diagram of phonon field resonance control.
图 5 金刚石第一布里渊区特征 (a)不含NV色心; (b)含NV色心
Figure 5. Characteristics of first Brillouin zone of diamond: (a) Without NV center; (b) contain NV center.
图 6 不同轴向NV色心金刚石的晶格能优化特征
Figure 6. Lattice energy optimization characteristics for the diamond with NV centers of different axes.
图 7 不同轴向NV色心金刚石的带隙特征 (a)无NV色心; (b) [1, 1, 1]轴向; (c) [1, –1, –1]轴向; (d) [–1, 1, –1]轴向; (e) [–1, –1, 1]轴向
Figure 7. Band gap characteristics for the diamond with NV centers of different axes: (a) Without NV center; (b) axis direction of [1, 1, 1]; (c) axis direction of [1, –1, –1]; (d) axis direction of [–1, 1, –1]; (e) axis direction of [–1, –1, 1].
图 8 不同轴向NV色心金刚石的态密度曲线
Figure 8. State density curves of the diamond with NV centers of different axes.
图 9 不同轴向NV色心金刚石的声子谱 (a)无NV色心; (b) [1, 1, 1]轴向; (c) [1, –1, –1]轴向; (d) [–1, 1, –1]轴向; (e) [–1, –1, 1]轴向
Figure 9. Phonon spectrum curves of the diamond with NV centers of different axes: (a) Without NV center; (b) axis direction of [1, 1, 1]; (c) axis direction of [1, –1, –1]; (d) axis direction of [–1, 1, –1]; (e) axis direction of [–1, –1, 1].
图 10 不同轴向NV色心金刚石的声子态密度曲线
Figure 10. Phonon state density curves of the diamond with NV centers of different axes.
图 11 不同轴向NV色心金刚石的Debye温度特征 (a)特征曲线; (b)特征值
Figure 11. Debye temperture characteristics of the diamond with NV centers of different axes: (a) Characteristic curves; (b) characteristic values.
图 12 不同轴向NV色心金刚石的声子热力学曲线 (a)热力学晗; (b)热力学熵; (c)热力学自由能
Figure 12. Debye temperture curves of the diamond with NV centers of different axes: (a) Enthalpy; (b) entropy; (c) free Energy.
图 13 不同轴向NV色心金刚石的热容特性 (a)热容曲线; (b)热容值
Figure 13. Heat capacity characteristics of the diamond with NV centers of different axes: (a) Heat capacity curves; (b) heat capacity values.
表 1 不同轴向NV色心的晶格动力学矩阵元的不对称关系
Table 1. Asymmetrical relations of lattice dynamics matrix elements for NV centers of different axes.
NV色心轴向 晶格动力学矩阵元不对称关系 NV色心轴向 晶格动力学矩阵元不对称关系 无NV色心 $\left\{ \begin{aligned}&{ {D_{xy} }\left( {{q} } \right) = {D_{yx} }\left( {{q} } \right)}\\&{ {D_{yz} }\left( {{q} } \right) = {D_{zy} }\left( {{q} } \right)}\\&{ {D_{xz} }\left( {{q} } \right) = {D_{zx} }\left( {{q} } \right)}\end{aligned} \right.$ [–1, 1, –1]轴向 $\left\{ {\begin{aligned}&{{D_{xy}}\left( {{q}} \right) = - {k_{[ - 1, 1, - 1]}}{D_{yx}}\left( {{q}} \right)}\\&{{D_{yz}}\left( {{q}} \right) = - {k_{[ - 1, 1, - 1]}}{D_{zy}}\left( {{q}} \right)}\\&{{D_{xz}}\left( {{q}} \right) = {k_{[ - 1, 1, - 1]}}{D_{zx}}\left( {{q}} \right)}\end{aligned}} \right.$ [1, 1, 1]轴向 $\left\{ {\begin{aligned}&{{D_{xy}}\left( {{q}} \right) = {k_{[1, 1, 1]}}{D_{yx}}\left( {{q}} \right)}\\&{{D_{yz}}\left( {{q}} \right) = {k_{[1, 1, 1]}}{D_{zy}}\left( {{q}} \right)}\\&{{D_{xz}}\left( {{q}} \right) = {k_{[1, 1, 1]}}{D_{zx}}\left( {{q}} \right)}\end{aligned}} \right.$ [–1, –1, 1]轴向 $\left\{ {\begin{aligned}&{{D_{xy}}\left( {{q}} \right) = {k_{[ - 1, - 1, 1]}}{D_{yx}}\left( {{q}} \right)}\\&{{D_{yz}}\left( {{q}} \right) = - {k_{[ - 1, - 1, 1]}}{D_{zy}}\left( {{q}} \right)}\\&{{D_{xz}}\left( {{q}} \right) = - {k_{[ - 1, - 1, 1]}}{D_{zx}}\left( {{q}} \right)}\end{aligned}} \right.$ [1, –1, –1]轴向 $\left\{ {\begin{aligned}&{{D_{xy}}\left( {{q}} \right) = - {k_{[1, - 1, - 1]}}{D_{yx}}\left( {{q}} \right)}\\&{{D_{yz}}\left( {{q}} \right) = {k_{[1, - 1, - 1]}}{D_{zy}}\left( {{q}} \right)}\\&{{D_{xz}}\left( {{q}} \right) = - {k_{[1, - 1, - 1]}}{D_{zx}}\left( {{q}} \right)}\end{aligned}} \right.$ 
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表 2 [1, 1, 1]轴向NV色心金刚石布里渊区特征线的声子谱解析结果
Table 2. Phonon spectrum analysis results at the characteristic line of the Brillouin zone in the diamond with the NV center of [1, 1, 1] axis.
特征线 声子谱波矢条件 声子谱函数 极化向量 Λ 线 $ {{q}}_{{x}}={{q}}_{y}={{q}}_{{z}}={q} $ $\left\{\begin{aligned}&{\omega }_{1}=\sqrt {{ {A} }_ {[1, 1, 1]} ^ {\varLambda } + {2}{B} _ {[1, 1, 1]} ^ {\varLambda }} \\ &{\omega }_{2}=\sqrt {{ {A} }_ {[1, 1, 1]} ^ {\varLambda } {-}{ {B} }_ {[1, 1, 1]} ^ {\varLambda } } \\ &{\omega }_{3}=\sqrt{ { {A} }_ {[1, 1, 1]} ^ {\varLambda } {-}{ {B} }_ {[1, 1, 1]} ^ {\varLambda } }\end{aligned}\right.$ $ \left\{\begin{aligned}&{{e}}_{{q}{1}}=\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)\\ &{{e}}_{{q}{2}}=\left({-}\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}{, 0}\right)\\ &{{e}}_{{q}{3}}=\left({-}\frac{1}{\sqrt{{6}}}{, -}\frac{1}{\sqrt{{6}}}, \frac{\sqrt{{6}}}{3}\right)\end{aligned}\right. $ $ \varDelta $线
(ΓF 线)
(ZQ 线)$ {{q}}_{{x}}={{q}}_{{z}}{=0} $ $\left\{\begin{aligned}&{\omega }_{1}=\sqrt{ { {A} }_{[1, 1, 1]}^{\varDelta }+{ {B} }_{[1, 1, 1]}^{\varDelta} }\\ &{\omega }_{2}=\sqrt{ { {B} }_{[1, 1, 1]}^{\varDelta } }\\ &{\omega }_{3}=\sqrt{ { {B} }_{[1, 1, 1]}^{\varDelta} }\end{aligned}\right.$ $ \left\{\begin{aligned}&{{e}}_{{q}{1}}=\left({0, 1, 0}\right)\\ &{{e}}_{{q}{2}}=\left({1, 0, 0}\right)\\ &{{e}}_{{q}{3}}=\left({0, 0, 1}\right)\end{aligned}\right. $ Σ 线 ${ {q} }_{ {x} }={ {q} }_{y}={q},$
$ {{q}}_{{z}}= 0 $$\left\{\begin{aligned}&{\omega }_{1}=\sqrt{ { {A} }_{ [1, 1, 1] }^{\varSigma }+{ {B} }_ {[1, 1, 1]} ^ {\varSigma } }\\ &{\omega }_{2}=\sqrt{ { {A} }_{[1, 1, 1]} ^ {\varSigma } {-}{ {B} }_{[1, 1, 1]} ^ {\varSigma } } \\ &{\omega }_{3}=\sqrt{ { {C} }_ {[1, 1, 1]} ^{\varSigma } } \end{aligned}\right.$ $ \left\{\begin{aligned}&{{e}}_{{q}{1}}=\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}{, 0}\right)\\ &{{e}}_{{q}{2}}=\left({-}\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}{, 0}\right)\\ &{{e}}_{{q}{3}}=\left({0, 0, 1}\right)\end{aligned}\right. $ M 线
(ΓZ 线)
(FQ 线)$ {{q}}_{{x}}={{q}}_{y}={0} $ $\left\{\begin{aligned}&{\omega }_{1}=\sqrt{ { {A} }_ {[1, 1, 1]} ^{ {M} }+{ {B} }_ {[1, 1, 1]} ^{ {M} } }\\ &{\omega }_{2}=\sqrt{ { {B} }_ {[1, 1, 1]} ^{ {M} } }\\ &{\omega }_{3}=\sqrt{ { {B} }_ {[1, 1, 1]} ^{ {M} } }\end{aligned}\right.$ $ \left\{\begin{aligned}&{{e}}_{{q}{1}}=\left({0, 0, 1}\right)\\ &{{e}}_{{q}{2}}=\left({1, 0, 0}\right)\\ &{{e}}_{{q}{3}}=\left({0, 1, 0}\right)\end{aligned}\right. $ 注: $A_{[1, 1, 1]}^\varDelta = \left( {2{f_1}/3{k_{[1, 1, 1]}}M_l^\alpha } \right)\left[ {2 - 2\cos \left( {{q_y}a/2} \right)} \right]$, $B_{[1, 1, 1]}^\varDelta = \left( {2{f_1}/3{k_{[1, 1, 1]} }M_l^\alpha } \right)\left[ {\eta - \eta \cos \left( { {q_y}a} \right)} \right]$,
$A_{[1, 1, 1]}^\varSigma = \left( { {f_1}/3{k_{[1, 1, 1]} }M_l^\alpha } \right)\{ 3 - 2\cos \left( {qa/2} \right) - \cos \left( {qa} \right) + \left[ {2\eta - 2\eta \cos \left( {qa} \right)} \right]\}$, $B_{[1, 1, 1]}^\varSigma = \left( { {f_1}/3{k_{[1, 1, 1]} }M_l^\alpha } \right)\left[ {1 - \cos \left( {qa} \right)} \right]$,
$C_{[1, 1, 1]}^\varSigma = \left( {2{f_1}/3{k_{[1, 1, 1]} }M_l^\alpha } \right)\left[ {2 - 2\cos \left( {qa/2} \right)} \right]$, $A_{[1, 1, 1]}^M = \left( {2{f_1}/3{k_{[1, 1, 1]}}M_l^\alpha } \right)\left[ {2 - 2\cos \left( {{q_z}a/2} \right)} \right]$,
$B_{[1, 1, 1]}^M = \left( {2{f_1}/3{k_{[1, 1, 1]}}M_l^\alpha } \right)\left[ {\eta - \eta \cos \left( {{q_z}a} \right)} \right]$.
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表 3 [1, 1, 1]轴向NV色心金刚石的声子热平衡温度解析结果
Table 3. Phonon thermal equilibrium temperature analysis results of the diamond with the NV center of [1, 1, 1] axis.
声子极化方向 声子热平衡温度 声子极化方向 声子热平衡温度 $ {\varLambda } $线方向 ${T}_{ {\varLambda } }=\dfrac{ {-}{\hbar }\sqrt{ { {A} }_{ {[1, 1, 1]} }^{ {\varLambda } }+{ {2}{B} }_{ {[1, 1, 1]} }^{ {\varLambda } } } }{ {k}_{\rm{B} }{\ln}\left(\frac{\left\langle { {n} } \right\rangle}{ {1+}\left\langle { {n} } \right\rangle}\right)}$ $ {\varSigma } $线方向 ${T}_{ {\varSigma } }=\dfrac{ {-}{\hbar }\sqrt{ { {A} }_{ {[1, 1, 1]} }^{ {\varSigma } }+{ {B} }_{ {[1, 1, 1]} }^{ {\varSigma } } } }{ {k}_{\rm{B} }{\ln}\left(\frac{\left\langle { {n} } \right\rangle}{ {1+}\left\langle { {n} } \right\rangle}\right)}$ $ \varDelta $线方向 ${T}_{\varDelta }=\dfrac{ {-}{\hbar }\sqrt{ { {A} }_{ {[1, 1, 1]} }^{\varDelta }+{ {B} }_{ {[1, 1, 1]} }^{\varDelta } } }{ {k}_{\rm{B} }{\ln}\left(\frac{\left\langle { {n} } \right\rangle}{ {1+}\left\langle { {n} } \right\rangle}\right)}$ M 线方向 ${T}_{ {M} }=\dfrac{ {-}{\hbar }\sqrt{ { {A} }_{ {[1, 1, 1]} }^{ {M} }+{ {B} }_{ {[1, 1, 1]} }^{ {M} } } }{ {k}_{\rm{B} }{\ln}\left(\frac{\left\langle { {n} } \right\rangle}{ {1+}\left\langle { {n} } \right\rangle}\right)}$ 注: 参数$ {{A}}_{{[1, 1, 1]}}^{{\varLambda }}, {{B}}_{{[1, 1, 1]}}^{{\varLambda }}, {{A}}_{{[1, 1, 1]}}^{\varDelta }, {{B}}_{{[1, 1, 1]}}^{\varDelta } $, $ {{A}}_{{[1, 1, 1]}}^{{\varSigma }}, {{B}}_{{[1, 1, 1]}}^{{\varSigma }}, {{A}}_{{[1, 1, 1]}}^{{M}} $和$ {{B}}_{{[1, 1, 1]}}^{{M}} $同表2.
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